Theorem ssrab 3345

Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Assertion

Ref	Expression
ssrab	|- (B C_ {x e. A | ph} <-> (B C_ A /\ A.x e. B ph))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   ph(x)

Proof of Theorem ssrab

Step	Hyp	Ref	Expression
1		df-rab 2624	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2	1	sseq2i 3297	. 2 |- (B C_ {x e. A | ph} <-> B C_ {x | (x e. A /\ ph)})
3		ssab 3337	. 2 |- (B C_ {x | (x e. A /\ ph)} <-> A.x(x e. B -> (x e. A /\ ph)))
4		dfss3 3264	. . . 4 |- (B C_ A <-> A.x e. B x e. A)
5	4	anbi1i 676	. . 3 |- ((B C_ A /\ A.x e. B ph) <-> (A.x e. B x e. A /\ A.x e. B ph))
6		r19.26 2747	. . 3 |- (A.x e. B (x e. A /\ ph) <-> (A.x e. B x e. A /\ A.x e. B ph))
7		df-ral 2620	. . 3 |- (A.x e. B (x e. A /\ ph) <-> A.x(x e. B -> (x e. A /\ ph)))
8	5, 6, 7	3bitr2ri 265	. 2 |- (A.x(x e. B -> (x e. A /\ ph)) <-> (B C_ A /\ A.x e. B ph))
9	2, 3, 8	3bitri 262	1 |- (B C_ {x e. A | ph} <-> (B C_ A /\ A.x e. B ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  {cab 2339  A.wral 2615  {crab 2619   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  ssrabdv  3346